Developing Mastery in Mathematics (1)

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Mastery in Maths: Research and lesson adaptation to fit the new criteria in Maths

An Action Research project by Clare Mondair

Aims

The aim of this investigation was to explore aspects of Mastery in Maths to improve my own understanding of what Mastery actually means and what it would mean for the students in my lessons. In addition, my aim was to change my own teaching where necessary in order to best help the students in my classes achieve of their best. As a Faculty we aimed to work together to develop lessons that would contain a ‘Mastery’ element as well as developing resources to add to the new Scheme of Work which was quite thin on the kind of Mastery resources required.

Literature Review

Although there are many differences between the education systems in England and Eastern Asia, the ‘mastery’ approach to teaching commonly followed in these countries can teach us much.

According to the National Centre for Excellence in the Teaching of Mathematics. (October 2014), the main principles and features characterised by this approach are that:

  • Teachers reinforce an expectation that all pupils are capable of achieving high standards in mathematics.
  • The large majority of pupils progress through the curriculum content at the same pace.
  • Differentiation is achieved by emphasising deep knowledge and through individual support and intervention.
  • Teaching is underpinned by methodical curriculum design and supported by carefully crafted lessons and resources to foster deep conceptual and procedural knowledge.
  • Practice and consolidation play a central role. Carefully designed variation within this builds fluency and understanding of underlying mathematical concepts in tandem.
  • Teachers use precise questioning in class to test conceptual and procedural knowledge, and assess pupils regularly to identify those requiring intervention so that all pupils keep up.

The intention of these approaches is to provide all children with full access to the curriculum, enabling them to achieve confidence and competence – ‘mastery’ – in mathematics, rather than many failing to develop the maths skills they need for the future. In addition, it has been recognised that for many schools and teachers the shift to this ‘mastery curriculum’ will be a significant one requiring new approaches to lesson design, teaching, use of resources and support of students. It also focuses on giving students the skills they need to make sensible choices and to use their knowledge to tackle problems in all sorts of situations. It also aims to develop their resilience for when the road gets tough.

There is a real need for a balanced approach here.  Of course having key facts at your disposal is very helpful when it comes to solving problems, especially when in an unfamiliar context but the need to be flexible and adaptable is too. Also helpful is being able to use what you do know to get you facts you don’t know.  For some students when learning their tables it can take a while to get 11 x 12 to “stick”, but if they are confident with 10 x 12 = 120 then they know how to get to the answer pretty quickly by adding on another 12.  Therefore, it is their understanding of the structure and not just my knowledge of the facts that helps out in tables tests. This is what sets students out on the road to mastery of the times tables!

Mastery of the Mathematics curriculum encourages ‘intelligent practice’ to enable students to develop conceptual understanding alongside procedural fluency.  It is important to use multiple representations to support this understanding and to encourage students’ reasoning.  Students are also encouraged to solve problems from the very start of the curriculum journey, not seeing it as some ‘add on’ that can only be considered when all the facts are in place.  The challenge is developing these skills and understanding alongside mastering different aspects of the Mathematics curriculum.

Asking students to think of more than one way to answer a question not only forces them to think more laterally but it also allows discussion of methods of true ’mastery’. These types of tasks enable students to:

  • Develop mathematical language.
  • Articulate their reasoning.
  • Share ideas on approaches to problem solving.
  • Grow in confidence when discussing ideas.

The key to this success is strong peer support which must be built up over time. In addition a good pedagogic tool to use in mathematical problem solving is instead of finding one way to solve a problem find three ways. Working in pairs is key to problem solving tasks as students come up with different ways of starting and after establishing one solution they are able to share alternative ideas. A core value of ‘mastery’ is partnership, listening to each other and showing respect for different views and ideas. Allowing students to explain their thinking, asking for and giving support and encouraging feedback is very important to establish and maintain mastery of mathematics.

Research Methods

Having researched and established as a Faculty what ‘mastery’ actually meant in maths we had to think about the way in which it would be developed. An agreement was made that we had already been attempting to develop mastery in terms of problem solving over the last year but lessons were intermittent and often lost out on due to other factors of school life.

An agreement was made that we would continue to develop lessons as normal but that a majority of our lessons would now contain an aspect of mastery. Nrich lessons would continue but would be incorporated in to the new Scheme of Work where possible (Nrich is a website created by Cambridge University which has open ended questions and what we now recognise as ‘Mastery challenges’). In addition, teachers in the Faculty would make lessons that they had produced available for the whole team and we would observe mastery lessons being delivered so that we would have a good understanding of what it ‘looked’ like. This would enable consistency throughout the Faculty. Many lessons would contain the format below so the students would know that was Nrich.

nrich-cm

The lesson would then progress through a series of steps so that there was enough challenge for everyone. Students were encouraged to work at their own level, however they were also encouraged to go for the higher level challenge where possible. This would result in those students who were less able working alongside those who already had a good ‘mastery’ of mathematics. The advantage of this system meant that students were working with and supporting peers and learning from one another in their language and at their own level.

find-a-whole-cm

The above would not apply to Year 11, but all other year groups should benefit from this thinking. It would of course be especially important for the Year 10 classes as they would be sitting a new exam which would call for resilience as well as thinking outside of the box in order to tackle some of the new material.

Main Findings

Asking students to think of more than one way to answer a question not only forces them to think more laterally but it also allows discussion of methods of true ’mastery’. These types of tasks enable students to:

  • Develop mathematical language.
  • Articulate their reasoning.
  • Share ideas on approaches to problem solving.
  • Grow in confidence when discussing ideas.

The key to this success is strong peer support which must be built up over time. In addition a good pedagogic tool to use in mathematical problem solving is instead of finding one way to solve a problem find three ways. Working in pairs is key to problem solving tasks as students come up with different ways of starting and after establishing one solution they are able to share alternative ideas. A core value of ‘mastery’ is partnership, listening to each other and showing respect for different views and ideas. Allowing students to explain their thinking, asking for and giving support and encouraging feedback is very important to establish and maintain mastery of mathematics.

Using Nrich we were able to appreciate that the current mastery approach encompasses two key aspects of mathematical learning, conceptual understanding and procedural fluency, which we agree are essential for nurturing young mathematicians. In addition there are five aspects of being able to be a Master at Maths, conceptual understanding; procedural fluency; strategic competence; adaptive reasoning and productive disposition (Kilpatrick, Stafford & Findell, 2001).

  • Much of the curriculum has been moved from higher levels to lower levels resulting in students now being expected to suddenly achieve at a much higher level than previously expected.
  • Many of our students do not have the basic mathematical fluency or reasoning skills in order to access much of the new curriculum.
  • Resilience in students is key in helping to ensure that students stay on track and improve.
  • Peer support and discussion is vital if students are to succeed in mastering some of the problem solving activities and questions which will come with the new curriculum.
  • Nrich allows students to explore Mathematics in a safe environment where they don’t feel threatened by their lack of basic knowledge.

Discussion and Conclusion

Mastery can only be developed over time and is unlikely to have much impact for the first two years of the new curriculum changes. The current difficulty we face is the fact that our Ks4 students have not been brought up with this habit of mastering Mathematics and it is therefore difficult to develop these skills and follow a Scheme of Work designed for a new exam which is already challenging to our average ability and less able students.

The mastery of Mathematics is however, being thoroughly embedded in the curriculum where possible for the Ks3 students and the impact of this should be felt when the current Year 9 group begin the GCSE course.

Mastery in Mathematics will enable students to articulate their ideas, build resilience, build mathematical fluency and think about problems from a different angle which in turn should have an impact on many aspects of life as well as Mathematics.

References

Drury, H. (2014) Mastering Mathematics. Oxford University Press, pp8.

Mathematics Learning Study Committee. Adding it up: Helping children learn mathematics. National Academies Press, 2001.

National Centre for Excellence in the Teaching of Mathematics. October 2014.

Williams, H. (2014) Approach, Research. Mathematics Mastery Acting Director of Primary.

 

 

 

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